A stable and efficient element-free Galerkin method is proposed for steady convection-diffusion problems.In the method integrations are computed with a local Taylor expansion nodal integral technique.According to convection-dominated degree,integration nodes are adaptively shifted opposite to the streamline direction.Compared with conventional element-free Galerkin method with stabilization,the method exhibits better stability and higher efficiency in solving convection-dominated convection-diffusion problems.It is a pure meshfree method,which is independent of background integral.Moreover,the method is easy to be implemented.

Velocity filed is decomposed into "coarse" and "fine" scales with a variational mulitiscale method.The "fine" scale is modeled by bubble functions,and solved with Petrov-Galerkin method.A stabilized term and stabilization parameter are introduced by coupling the "fine" and "coarse" scales.A variational multiscale equation which preserves properties of both "fine" and "coarse" scales is solved with a finite element method.It shows that the method is stable and accurate.It eliminates spurious oscillations caused by dominated advection term and uncoupling between velocity and pressure in numerical simulation of incompressible flows.The stabilization parameter can be applied to structure and unstructure meshes as well.

The buoyancy convective flows in rectangular cross-section cavity of porous media saturated with fluid and heated form below are studied.The finite difference methods are used to study the bifuraction structure based on the bifurcation theory.Some typical figures of streamline and isotherm are given.The model exchange mechanisms of convection are discussed.Numenical results indude critical Rayleigh numbers and critical aspect ratios for exchange of some lower modes.